Blind multiband signal reconstruction is the reconstruction of the multiband signal from sampling points in the time domain. The signal was discretized over an appropriately large frequency range containing all of its bands, and then signal reconstruction in frequency domain boils down to sparse signal recovery. Based on the fact that fewer sampling points were needed for compressed sensing recovery and the RIC (restricted isometry constant) of the observation matrix was required to be small enough to recover the sparse signal, a reweighted method was proposed to improve the condition number of the observation matrix so as to improve its RIC, as well as a corresponding reweighted OMP (orthogonal matching pursuit) scheme for blind multiband signal reconstruction. The proposed method was also applicable to the recovery of general sparse signals. In simulations, for an appropriately large frequency range, an appropriately small discrete interval satisfying the reconstruction error range was taken. The simulation results verify that for blind multiband signal reconstruction and the recovery of general sparse signals, the reweighted OMP scheme has a higher efficient reconstruction rate than the OMP algorithm directly used under the same conditions.